Extracting and representing qualitative behaviors of complex systems in phase space. (English) Zbl 0810.58011
Summary: This paper presents a computational method for automatically analyzing qualitative behaviors of complex dynamical systems in phase space. To demonstrate this method, a program called MAPS has been constructed that understands qualitatively distinct features of a phase space and represents geometric information about these features in a dimension- independent description, using deep domain knowledge of dynamical systems theory. Given a dynamical system specified as a system of governing equations, MAPS incrementally extracts the qualitative information about the system in terms of a qualitative phase-space structure describing steady-state behaviors, stabilities, and transient properties. MAPS generates a high-level symbolic description of the system sensible to human beings and manipulable by other programs, through a combination of numerical, combinatorial, and geometric computations and spatial reasoning techniques. MAPS has successfully demonstrated its power in a difficult engineering domain of nonlinear control design.
MSC:
| 37E99 | Low-dimensional dynamical systems |
| 94A15 | Information theory (general) |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 70G10 | Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics |
| 37N99 | Applications of dynamical systems |
Keywords:
steady-state behaviour; stability; complex dynamical systems; phase space; geometric information; transient propertiesSoftware:
MACSYMAReferences:
| [1] | Abelson, H., Bifurcation Interpreter: a step towards the automatic analysis of dynamical systems, Int. J. Comput. Math. Appl., 20, 8 (1990) |
| [2] | Abelson, H.; Eisenberg, M.; Halfant, M.; Katzenelson, J.; Sacks, E.; Sussman, G. J.; Wisdom, J.; Yip, K., Intelligence in scientific computing, Commun. ACM, 32, 5 (1989) |
| [3] | Boissonnat, J.-D., Geometric structures for three-dimensional shape representation, ACM Trans. Graphics, 3, 4 (1984) |
| [4] | Bradley, E., Control algorithms for chaotic systems, (Proceedings First European Conference on Algebraic Computing in Control (1991)) · Zbl 0793.93018 |
| [5] | Bradley, E.; Zhao, F., Phase-space control system design, IEEE Control Syst., 13, 2, 39-47 (1993) |
| [6] | Chiang, H.; Hirsh, M. W.; Wu, F. F., Stability regions of nonlinear autonomous dynamical systems, IEEE Trans. Autom. Control, 33, 1 (1988) · Zbl 0639.93043 |
| [7] | Clinger, W., \(Revised^4\) report on the algorithmic language scheme, (AIM-848b (1991), MIT AI Lab: MIT AI Lab Cambridge, MA) |
| [8] | Genesio, R.; Tartaglia, M.; Vicino, A., On the estimation of asymptotic stability regions: state of the art and new proposals, IEEE Trans. Autom. Control, 30, 8, 747-755 (1985) · Zbl 0568.93054 |
| [9] | Guckenheimer, J.; Holmes, P., (Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer: Springer New York) · Zbl 0515.34001 |
| [10] | Hirsh, M. W.; Smale, S., (Differential Equations, Dynamical Systems and Linear Algebra (1974), Academic Press: Academic Press New York) · Zbl 0309.34001 |
| [11] | Hsu, C. S., (Cell-to-Cell Mapping (1987), Springer: Springer New York) · Zbl 0632.58002 |
| [12] | Hsu, C. S.; Guttalu, R. S., An unraveling algorithm for global analysis of dynamical systems: an application of cell-to-cell mappings, J. Appl. Mech., 47, 940-948 (1980) · Zbl 0452.58020 |
| [13] | Munkres, J. R., (Elements of Algebraic Topology (1984), Addison-Wesley: Addison-Wesley Reading, MA) · Zbl 0673.55001 |
| [14] | Nishida, T.; Mizutani, K.; Kubota, A.; Doshita, S., Automated phase portrait analysis by integrating qualitative and quantitative analysis, (Proceedings AAAI-91. Proceedings AAAI-91, Anaheim, CA (1991)) |
| [15] | Ogata, K., (Modern Control Engineering (1970), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ) |
| [16] | Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S., Geometry from a time series, Phys. Rev. Lett., 45, 712-716 (1980) |
| [17] | Palacios-Velez, O.; Renaud, B. C., A dynamic hierarchical algorithm for computing Delaunay triangulations and other closest-point problems, ACM Trans. Math. Softw., 16, 3, 275-292 (1990) · Zbl 0883.68114 |
| [18] | Parker, T. S.; Chua, L. O., (Practical Numerical Algorithms for Chaotic Systems (1989), Springer: Springer New York) · Zbl 0692.58001 |
| [19] | Poincaré, H., Mémoire sur les courbes définies par une équation différentielle, J. Math., 7, 375 (1881) · JFM 13.0591.01 |
| [20] | Preparata, F. P.; Shamos, M. I., (Computational Geometry (1985), Springer: Springer New York) · Zbl 0759.68037 |
| [21] | Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., (Numerical Recipes: the Art of Scientific Computing (1988), Cambridge University Press: Cambridge University Press Cambridge, England) · Zbl 0661.65001 |
| [22] | Rand, R. H., (Computer Algebra in Applied Mathematics: An Introduction to MACSYMA (1984), Pitman: Pitman Boston, MA) · Zbl 0583.68012 |
| [23] | Sacks, E., Automatic analysis of one-parameter planar ordinary differential equations by intelligent numerical simulation, Artif. Intell., 48, 27-56 (1991) · Zbl 0714.65068 |
| [24] | Smale, S., Differentiable dynamical systems, Bull. AMS, 73 (1967) · Zbl 0202.55202 |
| [25] | Stoker, J. J., (Nonlinear Vibrations (1950), Interscience: Interscience New York) · Zbl 0035.39603 |
| [26] | Takens, F., Detecting strange attractors in turbulence, (Lecture Notes in Mathematics (1980), Springer: Springer New York), 366-381 · Zbl 0513.58032 |
| [27] | Yip, K. M., (KAM: A System for Intelligently Guiding Numerical Experimentation by Computer (1991), MIT Press: MIT Press Cambridge, MA) |
| [28] | Zhao, F., Extracting and representing qualitative behaviors of complex systems in phase spaces, (Proceedings IJCAI-91. Proceedings IJCAI-91, Sydney, Australia (1991)) · Zbl 0752.68076 |
| [29] | Zhao, F., Phase space navigator: towards automating control synthesis in phase spaces for nonlinear control systems, (Proceedings Third IFAC International Workshop on AI in Real Time Control (1991), Pergamon Press: Pergamon Press Oxford, England), also: MIT AI Lab. Report MIT-AIM-1286, Cambridge, MA |
| [30] | Zhao, F., Automatic analysis and synthesis of controllers for dynamical systems based on phase space knowledge, Ph.D. Thesis, MIT, MIT AI Lab Tech. Report AI-TR-1385, Cambridge, MA (1992) |
| [31] | Zhao, F., Computational dynamics: modeling and visualizing trajectory flows in phase space, Ann. Math. Artif. Intell., 8, 3-4, 285-300 (1993) · Zbl 1047.93500 |
| [32] | Zhao, F.; Thornton, R., Automatic design of a Maglev controller in state space, (Proceedings 31st IEEE Conference on Decision and Control. Proceedings 31st IEEE Conference on Decision and Control, Tucson, AZ (1992)), 2562-2567, also: MIT AI Lab. Report MIT-AIM-1303, Cambridge, MA |
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